What Is The Solution To The System Of Equations Below? Y = − 3 X + 5 Y = -3x + 5 Y = − 3 X + 5 And Y = 4 X − 2 Y = 4x - 2 Y = 4 X − 2 A. (1, 2) B. (1, -18) C. (-1, 8) D. (-1, -6)

Introduction

In mathematics, a system of equations is a set of two or more equations that are solved simultaneously to find the values of the variables. In this article, we will explore the solution to a system of two linear equations in two variables. We will use the method of substitution to solve the system of equations.

The System of Equations

The system of equations is given by:

$$y = -3x + 5$$

$$y = 4x - 2$$

Step 1: Equate the Two Equations

To solve the system of equations, we need to equate the two equations. This means that we need to set the two equations equal to each other.

$$-3x + 5 = 4x - 2$$

Step 2: Solve for x

Now, we need to solve for x. We can do this by isolating x on one side of the equation.

$$-3x - 4x = -2 - 5$$

$$-7x = -7$$

Step 3: Solve for x (continued)

Now, we need to solve for x by dividing both sides of the equation by -7.

$$x = \frac{-7}{-7}$$

$$x = 1$$

Step 4: Substitute x into One of the Original Equations

Now that we have found the value of x, we can substitute it into one of the original equations to find the value of y.

$$y = -3(1) + 5$$

$$y = -3 + 5$$

$$y = 2$$

The Solution

Therefore, the solution to the system of equations is (1, 2).

Conclusion

In this article, we have explored the solution to a system of two linear equations in two variables. We used the method of substitution to solve the system of equations and found the solution to be (1, 2).

Why is this Solution Correct?

This solution is correct because it satisfies both of the original equations. When we substitute x = 1 and y = 2 into the first equation, we get:

$$2 = -3(1) + 5$$

$$2 = -3 + 5$$

$$2 = 2$$

This shows that the solution (1, 2) satisfies the first equation. When we substitute x = 1 and y = 2 into the second equation, we get:

$$2 = 4(1) - 2$$

$$2 = 4 - 2$$

$$2 = 2$$

This shows that the solution (1, 2) satisfies the second equation. Therefore, the solution (1, 2) is correct.

What are the Other Options?

The other options are (1, -18), (-1, 8), and (-1, -6). Let's see if these options satisfy both of the original equations.

Option A: (1, -18)

When we substitute x = 1 and y = -18 into the first equation, we get:

$$-18 = -3(1) + 5$$

$$-18 = -3 + 5$$

$$- = -18 + 2$$

This shows that the option (1, -18) does not satisfy the first equation.

Option B: (1, -18) (continued)

When we substitute x = 1 and y = -18 into the second equation, we get:

$$-18 = 4(1) - 2$$

$$-18 = 4 - 2$$

$$-18 = 2$$

This shows that the option (1, -18) does not satisfy the second equation.

Option C: (-1, 8)

When we substitute x = -1 and y = 8 into the first equation, we get:

$$8 = -3(-1) + 5$$

$$8 = 3 + 5$$

$$8 = 8$$

This shows that the option (-1, 8) satisfies the first equation. When we substitute x = -1 and y = 8 into the second equation, we get:

$$8 = 4(-1) - 2$$

$$8 = -4 - 2$$

$$8 = -6$$

This shows that the option (-1, 8) does not satisfy the second equation.

Option D: (-1, -6)

When we substitute x = -1 and y = -6 into the first equation, we get:

$$-6 = -3(-1) + 5$$

$$-6 = 3 + 5$$

$$-6 = 8$$

This shows that the option (-1, -6) does not satisfy the first equation.

When we substitute x = -1 and y = -6 into the second equation, we get:

$$-6 = 4(-1) - 2$$

$$-6 = -4 - 2$$

$$-6 = -6$$

This shows that the option (-1, -6) satisfies the second equation. However, we have already found that the option (-1, -6) does not satisfy the first equation.

Conclusion

In this article, we have explored the solution to a system of two linear equations in two variables. We used the method of substitution to solve the system of equations and found the solution to be (1, 2). We also examined the other options and found that they do not satisfy both of the original equations.

Final Answer

Q: What is a system of equations?

A: A system of equations is a set of two or more equations that are solved simultaneously to find the values of the variables.

Q: How do I solve a system of equations?

A: There are several methods to solve a system of equations, including the method of substitution, the method of elimination, and the graphing method. In this article, we used the method of substitution to solve the system of equations.

Q: What is the method of substitution?

A: The method of substitution is a method of solving a system of equations by substituting one equation into the other equation. This method is useful when one of the equations is already solved for one of the variables.

Q: How do I use the method of substitution?

A: To use the method of substitution, follow these steps:

1. Write down the two equations.
2. Solve one of the equations for one of the variables.
3. Substitute the expression from step 2 into the other equation.
4. Solve the resulting equation for the other variable.
5. Substitute the value of the other variable back into one of the original equations to find the value of the first variable.

Q: What is the method of elimination?

A: The method of elimination is a method of solving a system of equations by adding or subtracting the two equations to eliminate one of the variables.

Q: How do I use the method of elimination?

A: To use the method of elimination, follow these steps:

1. Write down the two equations.
2. Multiply both equations by necessary multiples such that the coefficients of one of the variables are the same in both equations.
3. Add or subtract the two equations to eliminate one of the variables.
4. Solve the resulting equation for the other variable.
5. Substitute the value of the other variable back into one of the original equations to find the value of the first variable.

Q: What is the graphing method?

A: The graphing method is a method of solving a system of equations by graphing the two equations on a coordinate plane and finding the point of intersection.

Q: How do I use the graphing method?

A: To use the graphing method, follow these steps:

1. Write down the two equations.
2. Graph the two equations on a coordinate plane.
3. Find the point of intersection of the two graphs.
4. The point of intersection is the solution to the system of equations.

Q: What are some common mistakes to avoid when solving a system of equations?

A: Some common mistakes to avoid when solving a system of equations include:

• Not following the order of operations when simplifying expressions.
• Not checking the solution to make sure it satisfies both equations.
• Not using the correct method to solve the system of equations.
• Not being careful when adding or subtracting fractions.

Q: How do I check my solution to a system of equations?

A: To check your solution to a system of equations, follow these steps:

1. Substitute the values of the variables back into equations.
2. Simplify the expressions and check if they are true.
3. If the expressions are true, then the solution is correct.
4. If the expressions are not true, then the solution is incorrect.

Q: What are some real-world applications of systems of equations?

A: Some real-world applications of systems of equations include:

• Finding the cost and revenue of a business.
• Determining the amount of money in a bank account.
• Calculating the interest rate on a loan.
• Finding the distance and speed of an object.

Q: How do I use systems of equations in real-world applications?

A: To use systems of equations in real-world applications, follow these steps:

1. Identify the variables and the equations that relate them.
2. Write down the system of equations.
3. Solve the system of equations using the method of substitution, elimination, or graphing.
4. Use the solution to make decisions or predictions.

Q: What are some common types of systems of equations?

A: Some common types of systems of equations include:

• Linear systems of equations.
• Nonlinear systems of equations.
• Homogeneous systems of equations.
• Inhomogeneous systems of equations.

Q: How do I solve a linear system of equations?

A: To solve a linear system of equations, follow these steps:

1. Write down the two equations.
2. Use the method of substitution or elimination to solve the system of equations.
3. Check the solution to make sure it satisfies both equations.

Q: How do I solve a nonlinear system of equations?

A: To solve a nonlinear system of equations, follow these steps:

1. Write down the two equations.
2. Use numerical methods or graphical methods to solve the system of equations.
3. Check the solution to make sure it satisfies both equations.

Q: What are some common challenges when solving systems of equations?

A: Some common challenges when solving systems of equations include:

• Having multiple solutions or no solution.
• Having a system of equations that is inconsistent.
• Having a system of equations that is dependent.
• Having a system of equations that is inconsistent with the given information.

Q: How do I overcome these challenges?

A: To overcome these challenges, follow these steps:

1. Check the system of equations for consistency.
2. Check the system of equations for dependence.
3. Use numerical methods or graphical methods to solve the system of equations.
4. Check the solution to make sure it satisfies both equations.